987.231
Of the seven equatorial symmetries first employed in
the progression of self-
fractionations or cleavages, we use the tetrahedron's
six mid-edge poles to serve as the
three axes of spinnability. These three great-circle
spinnings delineate the succession of
cleavages of the 12 edges of the tetra-contained octahedron
whose six vertexes are
congruent with the regular tetrahedron's six midedge
polar spin points. The octahedron
resulting from the first cleavage has 12 edges; they
produce the additional external surface
lines necessary to describe the two-frequency, non-time-size
subdividing of the primitive
one-frequency tetrahedron. (See Sec.
526.23, which describes
how four happenings' loci
are required to produce and confirm a system discovery.)

987.232
The midpoints of the 12 edges of the octahedron formed
by the first cleavage
provide the 12 poles for the further great-circle spinning
and Cleavage #2 of both the tetra
and its contained octa by the six great circles of Symmetry
#3. Cleavage #2 also locates
the center-of-volume nucleus of the tetra and separates
out the center-of-volume-
surrounding 24 A Quanta Modules of the tetra and the
48 B Quanta Modules of the two-
frequency, tetra-contained octa. (See Sec.
942
for orientations
of the A and B Quanta
Modules.)

987.241
Symmetry #3 and Cleavage #3 mutually employ the six-polar-paired,
12
midedge points of the tetra-contained octa to produce
the six sets of great-circle
spinnabilities that in turn combine to define the two
(one positive, one negative) tetrahedra
that are intersymmetrically arrayed with the common-nuclear-vertexed
location of their
eight equi-interdistanced, outwardly and symmetrically
interarrayed vertexes of the
"cube"__the otherwise nonexistent, symmetric, square-windowed
hexahedron whose
overall most economical intervertexial relationship
lines are by themselves unstructurally
(nontriangularly) stabilized. The positive and negative
tetrahedra are internally trussed to
form a stable eight-cornered structure superficially
delineating a "cube" by the most
economical and intersymmetrical interrelationships of
the eight vertexes involved. (See
Fig.
987.240.)

987.242
In this positive-negative superficial cube of tetravolume-3
there is combined
an eight-faceted, asymmetric hourglass polyhedron of
tetravolume-l½, which
occurs interiorly of the interacting tetrahedra's edge
lines, and a complex asymmetric
doughnut cored hexahedron of tetravolume 1½, which surrounds
the interior
tetra's edge lines but occurs entirely inside and completely
fills the space between the
superficially described "cube" defined by the most economical
interconnecting of the eight
vertexes and the interior 1½-tetravolume hourglass core.
(See Fig. 987.242E987.242E.)

987.251
An example of Symmetry #4 appears at Fig.
450.10. An
example of
Symmetry #5 appears at Fig.
458.12B.
An example of Symmetry #6 appears at Fig.
458.12A.
An example of Symmetry #7 appears at Fig.
455.20.

987.300 Interactions of Symmetries: Spheric Domains

987.310 Irrationality of Nucleated and Nonnucleated Systems

987.311
The six great circles of Symmetry #3 interact with
the three great circles of
Symmetry # 1 to produce the 48 similar-surface triangles
ADH and AIH at Fig.
987.21ON.
The 48 similar triangles (24 plus, 24 minus) are
the surface-system set of the
48 similar asymmetric tetrahedra whose 48 central vertexes
are congruent in the
one__VE's__nuclear
vertex's center of volume.

987.312
These 48 asymmetric tetrahedra combine themselves into
12 sets of four
asymmetric tetra each. These 12 sets of four similar
(two positive, two negative)
asymmetric tetrahedra combine to define the 12 diamond
facets of the rhombic
dodecahedron of tetravolume-6. This rhombic dodecahedron's
hierarchical significance is
elsewhere identified as the allspace-filling domain
of each closest-packed, unit-radius
sphere in all isotropic, closest-packed, unit-radius
sphere aggregates, as the rhombic
dodecahedron's domain embraces both the unit-radius
sphere and that sphere's rationally
and exactly equal share of the intervening intersphere
space.

987.313
The four great circles of Symmetry #2 produce a minimum
nucleated system
of 12 unit-radius spheres closest packed tangentially
around each nuclear unit-radius
sphere; they also produce a polyhedral system of six
square windows and eight triangular
windows; they also produce four hexagonal planes of
symmetry that all pass through the
same nuclear vertex sphere's exact center.

987.314
These four interhexagonalling planes may also be seen
as the tetrahedron of
zero-time-size-volume because all of the latter's equi-edge
lengths, its face areas, and
system volumes are concurrently at zero.

987.315
This four-great-circle interaction in turn defines
the 24 equilengthed
vectorial radii and 24 equi-lengthed vector chords of
the VE. The 24 radii are grouped, by
construction, in two congruent sets, thereby to appear
as only 12 radii. Because the 24
radial vectors exactly equal energetically the circumferentially
closed system of 24
vectorial chords, we give this system the name vector
equilibrium. Its most unstable, only
transitional, equilibrious state serves nature's cosmic,
ceaseless, 100-percent-energy-
efficient, self-regenerative integrity by providing
the most expansive state of
intertransformation accommodation of the original hierarchy
of primitive, pre-time-size,
"click-stop" rational states of energy-involvement accountabilities.
Here we have in the
VE the eight possible phases of the initial positive-negative
tetrahedron occurring as an
inter-double-bonded (edge-bonded), vertex-paired, self-inter-coupling
nuclear system.

987.316
With the nucleated set of 12 equi-radius vertexial
spheres all closest packed
around one nuclear unit-radius sphere, we found we had
eight tetrahedra and six Half-
octahedra defined by this VE assembly, the total volume
of which is 20. But all of the six
Half-octahedra are completely unstable as the 12 spheres
cornering their six square
windows try to contract to produce six diamonds or 12
equiangular triangles to ensure
their interpatterning stability. (See Fig.
987.240.)

987.317
If we eliminate the nuclear sphere, the mass interattraction
of the 12
surrounding spheres immediately transforms their superficial
interpatterning into 20
equiangular triangles, and this altogether produces
the self-structuring pattern stability of
the 12 symmetrically interarrayed, but non-spherically-nucleated
icosahedron.

987.318
When this denucleation happens, the long diagonals
of the six squares
contract to unit-vector-radius length. The squares that
were enclosed on all four sides by
unit vectors were squares whose edges__being exactly
unity__had a diagonal hypotenuse
whose length was the second root of two__ergo, when VE
is transformed to the
icosahedron by the removal of the nuclear sphere, six
of its sqrt(2)-lengthed,
interattractive-relationship lines transform into a
length of 1, while the other 24 lines of
circumferential interattraction remain constant at unit-vector-radius
length. The difference
between the second root of two (which is 1.414214 -
1, i.e., the difference is 0.414214)
occurs six times, which amounts to a total system contraction
of 2.485284. This in turn
means that the original

24 + 8.485284 = 32.485284

overall unit-vector-lengths of containing bonds of the
VE are each reduced by a length of
2.485284 to an overall of exactly 30 unit-vector-radius
lengths.

987.319
This 2.485284 a excess of gravitational tensional-embracement
capability
constitutes the excess of intertransformative stretchability
between the VE's two
alternatively unstable, omnisystem's stable states and
its first two similarly stable,
omnitriangulated states.

987.320
Because the increment of instability tolerance of most
comprehensive
intertransformative events of the primitive hierarchy
is an irrational increment, the nucleus-
void icosahedron as a structural system is inherently
incommensurable with the nucleated
VE and its family of interrational values of the octahedral,
tetrahedral, and rhombic
dodecahedral states.

987.321
The irrational differences existing between nucleated
and nonnucleated
systems are probably the difference between proton-nucleated
and proton-neutron systems
and nonnucleated-nonneutroned electron systems, both
having identical numbers of
external closest-packed spheres, but having also different
overall, system-domain,
volumetric, and system-population involvements.

987.322
There is another important systemic difference between
VE's proton-neutron
system and the nonnucleated icosahedron's electron system:
the icosahedron is arrived at
by removing the nucleus, wherefore its contraction will
not permit the multilayering of
spheres as is permitted in the multilayerability of
the VE__ergo, it cannot have neutron
populating as in the VE; ergo, it permits only single-layer,
circumferential closest
packings; ergo, it permits only single spherical orbiting
domains of equal number to the
outer layers of VE-nucleated, closest-packed systems;
ergo, it permits only the behavioral
patterns of the electrons.

987.323
When all the foregoing is comprehended, it is realized
that the whole
concept of multiplication of information by division
also embraces the concept of
removing or separating out the nucleus sphere (vertex)
from the VE's structurally unstable
state and, as the jitterbug model shows, arriving omnisymmetrically
throughout the
transition at the structural stability of the icosahedron.
The icosahedron experimentally
evidences its further self-fractionation by its three
different polar great-circle hemispherical
cleavages that consistently follow the process of progressive
self-fractionations as spin-
halved successively around respective #5, #6, and #7
axes of symmetry. These successive
halvings develop various fractions corresponding in
arithmetical differentiation degrees, as
is shown in this exploratory accounting of the hierarchy
of unit-vector delineating
multiplication of information only by progressive subdividing
of parts.

987.324
When the tetrahedron is unity of tetravolume-1 (see
Table
223.64), then (in
contradistinction to the vector-radiused VE of tetravolume-20)

__

the vector-diametered VE = + 2½
or = - 2½

__

a rational, relative primitive prime number S tetravolume
is also only
realizable with half of its behavioral potentials in
the presently-tune-in-able
macrocosm and the other half of its total 5 behavioral
potential existent in the
presently-tune-out-able microcosm; thus,

__

an overall +5
tetravolume potential -2½__ergo, +5 - 2½ = +2½

or

__

an overall -5
tetravolume potential +2½ __ergo, - 5+2½ = - 2½

987.325
The positive and negative tetrahedra, when composited
as symmetrically
concentric and structurally stable, have eight symmetrically
interarranged vertexes defining
the corners of what in the past has been mistakenly
identified as a primitive polyhedron,
popularly and academically called the "cube" or hexahedron.
Cubes do not exist
primitively because they are structurally unstable,
having no triangularly-self-stabilizing
system pattern. They occur frequently in nature's crystals
but only as the superficial aspect
of a conglomerate complex of omnitriangulated polyhedra.

The three-great-circle symmetrical cleavaging (#I)
of the primitive tetrahedron
produces the vector-edged octahedron of tetravolume-4.

__

The vector-radiused rhombic triacontahedron, with
its .9994833324 unit-vector-
radius perpendicular to its midface center produces
a symmetrical polyhedron of
tetravolume-5.

__

With its 12 diamond-face-centers occurring at unit-vector-radius,
the rhombic
dodecahedron has a tetravolume-6.

The rhombic dodecahedron exactly occupies the geometric
domain of each unit-vector-
radius sphere and that sphere's external share of the
symmetrically identical spaces
intervening between closest-packed unit-radius spheres
of any and all aggregates of unit-
radius, closest-interpacked spheres. In this closest-packed
condition each sphere within the
aggregate always has 12 spheres symmetrically closest
packed tangentially around it. The
midpoints of the 12 diamond faces of the rhombic dodecahedron's
12 faces are congruent
with the points of tangency of the 12 surrounding spheres.
All the foregoing explains why
unit-radius rhombic dodecahedra fill allspace when joined
together.

987.327
Repeating the foregoing more economically we may say
that in this hierarchy
of omnisymmetric primitive polyhedra ranging from I
through 2, 2 , 3, 4, 5,
and 6 tetravolumes, the rhombic dodecahedron's 12 diamond-face-midpoints
occur at the
points of intertangency of the 12 surrounding spheres.
It is thus disclosed that the rhombic
dodecahedron is not only the symmetric domain of both
the sphere itself and the sphere's
symmetric share of the space intervening between all
closest-packed spheres and therefore
also of the nuclear domains of all isotropic vector
matrixes (Sec.
420), but the rhombic
dodecahedron is also the maximum-limit-volumed primitive
polyhedron of frequency-l.